And I don’t mean the Beatles or Abba.
The oldest example in my memory is something that went by the name of “Sukiyaki” in the English speaking world. It’s a wistful Japanese ballad called “Ue wo muite arukou” from 1961.
A bit later we have Hugh Masekela’s “Grazing in the Grass” from 1968. Though I guess that doesn’t count because, though Masekela is from South Africa, he was in the USA when he recorded it and it broke in the American market. But I’m including it here because of Masakela’s lineage (and because of the cowbell).
Still more recently there’s Israel Kamakawiwo'ole’s (aka IZ) mash-up of “What a Wonderful World” and “Over the Rainbow.” It went viral in 1993. Yes, Hawaii is an American state, but IZ was a native Hawaiian and recorded as such. He recomposed those tunes to fit his aesthetic.
The most recent examples I can think of are “Despacito” and “Dance Monkey”. “Despacito” is out of Puerto Rico (2017) by Luis Fonsi and Daddy Yankee; it later got a boost from a Justin Bieber remix. “Dance Monkey” came from Australia by Tones and I in 2019. I recently became aware of it through a performance by Kwak DaKyung, a young Korean trumpet prodigy, and Henry Lau, a K-pop star.
Other examples? “Macarena” from the mid-1990s? Surely there are examples from earlier in the 20th century. What about the 19th? Of course the mass media as we know it didn’t exist back then. But still, music gets around.
In particular, I’m thinking of a tune that goes by various names (e.g. “The Streets of Cairo,” “Arabian Song”), and was copyrighted under five different titles in America in the 1990s. It has been very tentatively traced back to the Arab world in the 17th century. At some point it began showing up accompanying snake charmers in cartoons. I first knew is as a children’s song, with slightly salacious lyrics, from my own mid-20th century childhood.